cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A369057 Total number of representations of natural numbers in range 1 .. 4n-1 as sums of the form p*q + p*r + q*r, with three odd primes p <= q <= r.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 4, 4, 4, 4, 5, 7, 7, 7, 8, 9, 10, 10, 11, 11, 12, 12, 14, 15, 15, 16, 17, 17, 17, 19, 21, 22, 22, 22, 24, 24, 24, 24, 25, 26, 28, 30, 31, 32, 32, 33, 35, 35, 35, 35, 38, 38, 39, 39, 41, 42, 44, 44, 47, 48, 49, 50, 50, 50, 51, 52, 52, 54, 54, 54, 59, 61, 61, 61, 63, 64, 65, 65, 67
Offset: 1

Views

Author

Antti Karttunen, Jan 21 2024

Keywords

Comments

Terms a(10^n), for n=1..7 are: 2, 82, 1819, 34220, 628914, 11855507, 233030075, which gives a(n)/n ratios: 0.2, 0.82, 1.82, 3.42, 6.29, 11.86, 23.30, etc, Question: does the ratio just keep on growing?

Links

Crossrefs

Partial sums of A369055.
Cf. A369054.

Programs

  • PARI
    \\ Needs also program from A369055:
A369057list(up_to) = { my(v=vector(up_to)); s = 0; for(n=1,up_to,s+=A369055(n); v[n] = s); (v); };
v369057 = A369057list(up_to);
A369057(n) = v369057[n];

A369242 Number of representations of n! - 1 as a sum (p*q + p*r + q*r) with three odd primes p <= q <= r.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 1, 3, 11, 24, 53, 176, 339, 1510, 2573
Offset: 0

Views

Author

Antti Karttunen, Jan 21 2024

Keywords

Crossrefs

Cf. A033312, A369054.
Cf. also A369241, A369245.

Programs

Formula

a(n) = A369054(A033312(n)).
For n >= 4, a(n) = A369055(n!/4).

A369463 Numbers of the form 12*m-1 for which there is no representation as a sum (p*q + p*r + q*r) with three odd primes p <= q <= r.

Original entry on oeis.org

11, 23, 35, 47, 59, 83, 107, 143, 179, 227, 323, 347, 443, 515, 659, 683, 827, 947, 1259, 1523, 1763, 1787, 2075, 2267, 2675, 2963, 3023, 3203, 3275, 3347, 3467, 3635, 4523, 4643, 4859, 5003, 5147, 5747, 5819, 6395, 6803, 6827, 7235, 8003, 8123, 8171, 8747, 8963, 9323, 9659, 9827, 10367, 10427, 12347, 12923, 13187
Offset: 1

Views

Author

Antti Karttunen, Jan 23 2024

Keywords

Comments

Equal to (12*i)-1, where i are the positions of 0's in A369462.
Terms of the form 3k+2 in A369056. These seem to be much more rare than terms of A369248.
Question: Is this a finite sequence, with the last term a(285) = 50688947 = (12*4224079)-1? See conjecture in A369055.
If it exists, a(286) > 201326603 (= (12*(2^24))+11).

Links

Crossrefs

Intersection of A016789 and A369056 (and of A369464).
Subsequence of A017653.
Cf. A369054, A369055, A369462.
Cf. also A369248.

Programs

  • PARI
    isA369251(n) =  if(3!=(n%4),0, my(v = [3,3], ip = #v, r); while(1, r = (n-(v[1]*v[2])) / (v[1]+v[2]); if(r < v[2], ip--, ip = #v; if(1==denominator(r) && isprime(r), return(1))); if(!ip, return(0)); v[ip] = nextprime(1+v[ip]); for(i=1+ip,#v,v[i]=v[i-1])));
isA369463(n) = ((11==(n%12)) && !isA369251(n));

A369064 Record values in A369054.

Original entry on oeis.org

0, 1, 2, 3, 5, 8, 10, 12, 13, 15, 24, 25, 31, 33, 34, 35, 40, 43, 44, 45, 46, 49, 51, 54, 57, 58, 63, 65, 67, 68, 69, 71, 76, 78, 79, 80, 81, 83, 84, 85, 87, 95, 99, 103, 105, 106, 109, 120, 125, 132, 136, 152, 153, 157, 159, 162, 166, 171, 178, 180, 181, 186, 198, 213, 217, 219, 226, 228, 229, 231, 238, 246, 261, 263, 264, 270, 296, 302, 310
Offset: 1

Views

Author

Antti Karttunen, Jan 21 2024

Keywords

Crossrefs

Also record values in A369055.
Cf. A369054, A369063.

Formula

a(n) = A369054(A369063(n)).
Previous Showing 11-14 of 14 results.

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